Given $X,Y$ Banach spaces and $T \in L(X,Y)$, show that the following sentences are equivalent:
A) there exists $S \in L(Y,X)$ such that $S(T(x))=x$ for all $x \in X$.
B) $T$ is injective and $T(X)$ is a complemented space of $Y$.
I was given this exercise in my functional analysis course but I don’t know how to solve this.
All I have understood so far in this exercise are the following:
- I was given the following definition of "complemented space'': a closed subspace $M$ is complemented in $N$ if exists a topological complement of $M$ in $N$ or equivalently if there exists a linear continuous projection $P$ in $N$ such that $𝑃(N)=M$;
- $L(X,Y)$ means the set of all continuous linear operators from $X$ to $Y$.